2009-12-16 18:17:34 +01:00
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function [dr,info,M_,options_,oo_] = dr11_sparse(dr,task,M_,options_,oo_, jacobia_, hessian)
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%function [dr,info,M_,options_,oo_] = dr11_sparse(dr,task,M_,options_,oo_, jacobia_, hessian)
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% Copyright (C) 2008 Dynare Team
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%
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% This file is part of Dynare.
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%
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% Dynare is free software: you can redistribute it and/or modify
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% it under the terms of the GNU General Public License as published by
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% the Free Software Foundation, either version 3 of the License, or
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% (at your option) any later version.
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%
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% Dynare is distributed in the hope that it will be useful,
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% but WITHOUT ANY WARRANTY; without even the implied warranty of
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% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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% GNU General Public License for more details.
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%
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% You should have received a copy of the GNU General Public License
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% along with Dynare. If not, see <http://www.gnu.org/licenses/>.
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%task
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info = 0;
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klen = M_.maximum_endo_lag + M_.maximum_endo_lead + 1;
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kstate = dr.kstate;
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kad = dr.kad;
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kae = dr.kae;
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nstatic = dr.nstatic;
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nfwrd = dr.nfwrd;
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npred = dr.npred;
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nboth = dr.nboth;
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order_var = dr.order_var;
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nd = size(kstate,1);
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nz = nnz(M_.lead_lag_incidence);
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sdyn = M_.endo_nbr - nstatic;
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k0 = M_.lead_lag_incidence(M_.maximum_endo_lag+1,order_var);
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k1 = M_.lead_lag_incidence(find([1:klen] ~= M_.maximum_endo_lag+1),:);
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b = jacobia_(:,k0);
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if M_.maximum_endo_lead == 0; % backward models
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a = jacobia_(:,nonzeros(k1'));
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dr.ghx = zeros(size(a));
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m = 0;
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for i=M_.maximum_endo_lag:-1:1
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k = nonzeros(M_.lead_lag_incidence(i,order_var));
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dr.ghx(:,m+[1:length(k)]) = -b\a(:,k);
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m = m+length(k);
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end
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if M_.exo_nbr & task~=1
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jacobia_
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jacobia_(:,nz+1:end)
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b
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dr.ghu = -b\jacobia_(:,nz+1:end);
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disp(['nz=' int2str(nz) ]);
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dr.ghu
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end
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dr.eigval = eig(transition_matrix(dr,M_));
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dr.rank = 0;
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if any(abs(dr.eigval) > options_.qz_criterium)
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temp = sort(abs(dr.eigval));
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nba = nnz(abs(dr.eigval) > options_.qz_criterium);
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temp = temp(nd-nba+1:nd)-1-options_.qz_criterium;
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info(1) = 3;
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info(2) = temp'*temp;
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end
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return;
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end
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%forward--looking models
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if nstatic > 0
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[Q,R] = qr(b(:,1:nstatic));
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aa = Q'*jacobia_;
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else
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aa = jacobia_;
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end
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a = aa(:,nonzeros(k1'));
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b = aa(:,k0);
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b10 = b(1:nstatic,1:nstatic);
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b11 = b(1:nstatic,nstatic+1:end);
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b2 = b(nstatic+1:end,nstatic+1:end);
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if any(isinf(a(:)))
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info = 1;
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return
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end
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% buildind D and E
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%nd
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d = zeros(nd,nd) ;
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e = d ;
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k = find(kstate(:,2) >= M_.maximum_endo_lag+2 & kstate(:,3));
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d(1:sdyn,k) = a(nstatic+1:end,kstate(k,3)) ;
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k1 = find(kstate(:,2) == M_.maximum_endo_lag+2);
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e(1:sdyn,k1) = -b2(:,kstate(k1,1)-nstatic);
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k = find(kstate(:,2) <= M_.maximum_endo_lag+1 & kstate(:,4));
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e(1:sdyn,k) = -a(nstatic+1:end,kstate(k,4)) ;
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k2 = find(kstate(:,2) == M_.maximum_endo_lag+1);
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k2 = k2(~ismember(kstate(k2,1),kstate(k1,1)));
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d(1:sdyn,k2) = b2(:,kstate(k2,1)-nstatic);
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if ~isempty(kad)
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for j = 1:size(kad,1)
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d(sdyn+j,kad(j)) = 1 ;
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e(sdyn+j,kae(j)) = 1 ;
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end
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end
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%e
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%d
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[ss,tt,w,sdim,dr.eigval,info1] = mjdgges(e,d,options_.qz_criterium);
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if info1
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info(1) = 2;
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info(2) = info1;
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return
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end
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nba = nd-sdim;
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nyf = sum(kstate(:,2) > M_.maximum_endo_lag+1);
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if task == 1
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dr.rank = rank(w(1:nyf,nd-nyf+1:end));
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% Under Octave, eig(A,B) doesn't exist, and
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% lambda = qz(A,B) won't return infinite eigenvalues
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if ~exist('OCTAVE_VERSION')
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dr.eigval = eig(e,d);
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end
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return
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end
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if nba ~= nyf
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temp = sort(abs(dr.eigval));
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if nba > nyf
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temp = temp(nd-nba+1:nd-nyf)-1-options_.qz_criterium;
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info(1) = 3;
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elseif nba < nyf;
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temp = temp(nd-nyf+1:nd-nba)-1-options_.qz_criterium;
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info(1) = 4;
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end
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info(2) = temp'*temp;
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return
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end
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np = nd - nyf;
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n2 = np + 1;
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n3 = nyf;
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n4 = n3 + 1;
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% derivatives with respect to dynamic state variables
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% forward variables
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w1 =w(1:n3,n2:nd);
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if condest(w1) > 1e9;
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info(1) = 5;
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info(2) = condest(w1);
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return;
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else
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gx = -w1'\w(n4:nd,n2:nd)';
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end
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% predetermined variables
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hx = w(1:n3,1:np)'*gx+w(n4:nd,1:np)';
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hx = (tt(1:np,1:np)*hx)\(ss(1:np,1:np)*hx);
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k1 = find(kstate(n4:nd,2) == M_.maximum_endo_lag+1);
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k2 = find(kstate(1:n3,2) == M_.maximum_endo_lag+2);
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hx(k1,:)
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gx(k2(nboth+1:end),:)
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dr.ghx = [hx(k1,:); gx(k2(nboth+1:end),:)];
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dr.ghx
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%lead variables actually present in the model
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j3 = nonzeros(kstate(:,3));
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j4 = find(kstate(:,3));
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% derivatives with respect to exogenous variables
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disp(['M_.exo_nbr=' int2str(M_.exo_nbr)]);
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if M_.exo_nbr
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fu = aa(:,nz+(1:M_.exo_nbr));
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a1 = b;
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aa1 = [];
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if nstatic > 0
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aa1 = a1(:,1:nstatic);
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end
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dr.ghu = -[aa1 a(:,j3)*gx(j4,1:npred)+a1(:,nstatic+1:nstatic+ ...
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npred) a1(:,nstatic+npred+1:end)]\fu;
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else
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dr.ghu = [];
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end
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% static variables
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if nstatic > 0
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temp = -a(1:nstatic,j3)*gx(j4,:)*hx;
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j5 = find(kstate(n4:nd,4));
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temp(:,j5) = temp(:,j5)-a(1:nstatic,nonzeros(kstate(:,4)));
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temp = b10\(temp-b11*dr.ghx);
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dr.ghx = [temp; dr.ghx];
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temp = [];
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end
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if options_.loglinear == 1
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k = find(dr.kstate(:,2) <= M_.maximum_endo_lag+1);
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klag = dr.kstate(k,[1 2]);
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k1 = dr.order_var;
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dr.ghx = repmat(1./dr.ys(k1),1,size(dr.ghx,2)).*dr.ghx.* ...
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repmat(dr.ys(k1(klag(:,1)))',size(dr.ghx,1),1);
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dr.ghu = repmat(1./dr.ys(k1),1,size(dr.ghu,2)).*dr.ghu;
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end
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%% Necessary when using Sims' routines for QZ
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if options_.use_qzdiv
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gx = real(gx);
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hx = real(hx);
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dr.ghx = real(dr.ghx);
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dr.ghu = real(dr.ghu);
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end
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%exogenous deterministic variables
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if M_.exo_det_nbr > 0
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f1 = sparse(jacobia_(:,nonzeros(M_.lead_lag_incidence(M_.maximum_endo_lag+2:end,order_var))));
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f0 = sparse(jacobia_(:,nonzeros(M_.lead_lag_incidence(M_.maximum_endo_lag+1,order_var))));
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fudet = sparse(jacobia_(:,nz+M_.exo_nbr+1:end));
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M1 = inv(f0+[zeros(M_.endo_nbr,nstatic) f1*gx zeros(M_.endo_nbr,nyf-nboth)]);
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M2 = M1*f1;
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dr.ghud = cell(M_.exo_det_length,1);
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dr.ghud{1} = -M1*fudet;
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for i = 2:M_.exo_det_length
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dr.ghud{i} = -M2*dr.ghud{i-1}(end-nyf+1:end,:);
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end
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end
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if options_.order == 1
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return
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end
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% Second order
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%tempex = oo_.exo_simul ;
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%hessian = real(hessext('ff1_',[z; oo_.exo_steady_state]))' ;
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kk = flipud(cumsum(flipud(M_.lead_lag_incidence(M_.maximum_endo_lag+1:end,order_var)),1));
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if M_.maximum_endo_lag > 0
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kk = [cumsum(M_.lead_lag_incidence(1:M_.maximum_endo_lag,order_var),1); kk];
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end
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kk = kk';
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kk = find(kk(:));
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nk = size(kk,1) + M_.exo_nbr + M_.exo_det_nbr;
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k1 = M_.lead_lag_incidence(:,order_var);
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k1 = k1';
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k1 = k1(:);
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k1 = k1(kk);
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k2 = find(k1);
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kk1(k1(k2)) = k2;
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kk1 = [kk1 length(k1)+1:length(k1)+M_.exo_nbr+M_.exo_det_nbr];
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kk = reshape([1:nk^2],nk,nk);
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kk1 = kk(kk1,kk1);
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%[junk,junk,hessian] = feval([M_.fname '_dynamic'],z, oo_.exo_steady_state);
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hessian(:,kk1(:)) = hessian;
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%oo_.exo_simul = tempex ;
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%clear tempex
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n1 = 0;
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n2 = np;
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zx = zeros(np,np);
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zu=zeros(np,M_.exo_nbr);
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for i=2:M_.maximum_endo_lag+1
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k1 = sum(kstate(:,2) == i);
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zx(n1+1:n1+k1,n2-k1+1:n2)=eye(k1);
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n1 = n1+k1;
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n2 = n2-k1;
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end
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kk = flipud(cumsum(flipud(M_.lead_lag_incidence(M_.maximum_endo_lag+1:end,order_var)),1));
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k0 = [1:M_.endo_nbr];
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gx1 = dr.ghx;
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hu = dr.ghu(nstatic+[1:npred],:);
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zx = [zx; gx1];
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zu = [zu; dr.ghu];
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for i=1:M_.maximum_endo_lead
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k1 = find(kk(i+1,k0) > 0);
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zu = [zu; gx1(k1,1:npred)*hu];
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gx1 = gx1(k1,:)*hx;
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zx = [zx; gx1];
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kk = kk(:,k0);
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k0 = k1;
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end
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zx=[zx; zeros(M_.exo_nbr,np);zeros(M_.exo_det_nbr,np)];
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zu=[zu; eye(M_.exo_nbr);zeros(M_.exo_det_nbr,M_.exo_nbr)];
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[nrzx,nczx] = size(zx);
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rhs = -sparse_hessian_times_B_kronecker_C(hessian,zx);
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%lhs
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n = M_.endo_nbr+sum(kstate(:,2) > M_.maximum_endo_lag+1 & kstate(:,2) < M_.maximum_endo_lag+M_.maximum_endo_lead+1);
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A = zeros(n,n);
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B = zeros(n,n);
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A(1:M_.endo_nbr,1:M_.endo_nbr) = jacobia_(:,M_.lead_lag_incidence(M_.maximum_endo_lag+1,order_var));
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% variables with the highest lead
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k1 = find(kstate(:,2) == M_.maximum_endo_lag+M_.maximum_endo_lead+1);
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if M_.maximum_endo_lead > 1
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k2 = find(kstate(:,2) == M_.maximum_endo_lag+M_.maximum_endo_lead);
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[junk,junk,k3] = intersect(kstate(k1,1),kstate(k2,1));
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else
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k2 = [1:M_.endo_nbr];
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k3 = kstate(k1,1);
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end
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% Jacobian with respect to the variables with the highest lead
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B(1:M_.endo_nbr,end-length(k2)+k3) = jacobia_(:,kstate(k1,3)+M_.endo_nbr);
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offset = M_.endo_nbr;
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k0 = [1:M_.endo_nbr];
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gx1 = dr.ghx;
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for i=1:M_.maximum_endo_lead-1
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k1 = find(kstate(:,2) == M_.maximum_endo_lag+i+1);
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[k2,junk,k3] = find(kstate(k1,3));
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A(1:M_.endo_nbr,offset+k2) = jacobia_(:,k3+M_.endo_nbr);
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n1 = length(k1);
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A(offset+[1:n1],nstatic+[1:npred]) = -gx1(kstate(k1,1),1:npred);
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gx1 = gx1*hx;
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A(offset+[1:n1],offset+[1:n1]) = eye(n1);
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n0 = length(k0);
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E = eye(n0);
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if i == 1
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[junk,junk,k4]=intersect(kstate(k1,1),[1:M_.endo_nbr]);
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else
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[junk,junk,k4]=intersect(kstate(k1,1),kstate(k0,1));
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end
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i1 = offset-n0+n1;
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B(offset+[1:n1],offset-n0+[1:n0]) = -E(k4,:);
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k0 = k1;
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offset = offset + n1;
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end
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[junk,k1,k2] = find(M_.lead_lag_incidence(M_.maximum_endo_lag+M_.maximum_endo_lead+1,order_var));
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A(1:M_.endo_nbr,nstatic+1:nstatic+npred)=...
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A(1:M_.endo_nbr,nstatic+[1:npred])+jacobia_(:,k2)*gx1(k1,1:npred);
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C = hx;
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D = [rhs; zeros(n-M_.endo_nbr,size(rhs,2))];
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dr.ghxx = gensylv(2,A,B,C,D);
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%ghxu
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%rhs
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hu = dr.ghu(nstatic+1:nstatic+npred,:);
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%kk = reshape([1:np*np],np,np);
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%kk = kk(1:npred,1:npred);
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%rhs = -hessian*kron(zx,zu)-f1*dr.ghxx(end-nyf+1:end,kk(:))*kron(hx(1:npred,:),hu(1:npred,:));
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rhs = sparse_hessian_times_B_kronecker_C(hessian,zx,zu);
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nyf1 = sum(kstate(:,2) == M_.maximum_endo_lag+2);
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hu1 = [hu;zeros(np-npred,M_.exo_nbr)];
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%B1 = [B(1:M_.endo_nbr,:);zeros(size(A,1)-M_.endo_nbr,size(B,2))];
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[nrhx,nchx] = size(hx);
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[nrhu1,nchu1] = size(hu1);
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B1 = B*A_times_B_kronecker_C(dr.ghxx,hx,hu1);
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rhs = -[rhs; zeros(n-M_.endo_nbr,size(rhs,2))]-B1;
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%lhs
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dr.ghxu = A\rhs;
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%ghuu
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%rhs
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kk = reshape([1:np*np],np,np);
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kk = kk(1:npred,1:npred);
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rhs = sparse_hessian_times_B_kronecker_C(hessian,zu);
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B1 = A_times_B_kronecker_C(B*dr.ghxx,hu1);
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|
rhs = -[rhs; zeros(n-M_.endo_nbr,size(rhs,2))]-B1;
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%lhs
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dr.ghuu = A\rhs;
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dr.ghxx = dr.ghxx(1:M_.endo_nbr,:);
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dr.ghxu = dr.ghxu(1:M_.endo_nbr,:);
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|
dr.ghuu = dr.ghuu(1:M_.endo_nbr,:);
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% dr.ghs2
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|
% derivatives of F with respect to forward variables
|
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|
% reordering predetermined variables in diminishing lag order
|
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|
O1 = zeros(M_.endo_nbr,nstatic);
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|
O2 = zeros(M_.endo_nbr,M_.endo_nbr-nstatic-npred);
|
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|
LHS = jacobia_(:,M_.lead_lag_incidence(M_.maximum_endo_lag+1,order_var));
|
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|
RHS = zeros(M_.endo_nbr,M_.exo_nbr^2);
|
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|
kk = find(kstate(:,2) == M_.maximum_endo_lag+2);
|
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|
gu = dr.ghu;
|
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|
guu = dr.ghuu;
|
|
|
|
Gu = [dr.ghu(nstatic+[1:npred],:); zeros(np-npred,M_.exo_nbr)];
|
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|
|
Guu = [dr.ghuu(nstatic+[1:npred],:); zeros(np-npred,M_.exo_nbr*M_.exo_nbr)];
|
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|
|
E = eye(M_.endo_nbr);
|
|
|
|
M_.lead_lag_incidenceordered = flipud(cumsum(flipud(M_.lead_lag_incidence(M_.maximum_endo_lag+1:end,order_var)),1));
|
|
|
|
if M_.maximum_endo_lag > 0
|
|
|
|
M_.lead_lag_incidenceordered = [cumsum(M_.lead_lag_incidence(1:M_.maximum_endo_lag,order_var),1); M_.lead_lag_incidenceordered];
|
|
|
|
end
|
|
|
|
M_.lead_lag_incidenceordered = M_.lead_lag_incidenceordered';
|
|
|
|
M_.lead_lag_incidenceordered = M_.lead_lag_incidenceordered(:);
|
|
|
|
k1 = find(M_.lead_lag_incidenceordered);
|
|
|
|
M_.lead_lag_incidenceordered(k1) = [1:length(k1)]';
|
|
|
|
M_.lead_lag_incidenceordered =reshape(M_.lead_lag_incidenceordered,M_.endo_nbr,M_.maximum_endo_lag+M_.maximum_endo_lead+1)';
|
|
|
|
kh = reshape([1:nk^2],nk,nk);
|
|
|
|
kp = sum(kstate(:,2) <= M_.maximum_endo_lag+1);
|
|
|
|
E1 = [eye(npred); zeros(kp-npred,npred)];
|
|
|
|
H = E1;
|
|
|
|
hxx = dr.ghxx(nstatic+[1:npred],:);
|
|
|
|
for i=1:M_.maximum_endo_lead
|
|
|
|
for j=i:M_.maximum_endo_lead
|
|
|
|
[junk,k2a,k2] = find(M_.lead_lag_incidence(M_.maximum_endo_lag+j+1,order_var));
|
|
|
|
[junk,k3a,k3] = ...
|
|
|
|
find(M_.lead_lag_incidenceordered(M_.maximum_endo_lag+j+1,:));
|
|
|
|
nk3a = length(k3a);
|
|
|
|
B1 = sparse_hessian_times_B_kronecker_C(hessian(:,kh(k3,k3)),gu(k3a,:));
|
|
|
|
RHS = RHS + jacobia_(:,k2)*guu(k2a,:)+B1;
|
|
|
|
end
|
|
|
|
% LHS
|
|
|
|
[junk,k2a,k2] = find(M_.lead_lag_incidence(M_.maximum_endo_lag+i+1,order_var));
|
|
|
|
LHS = LHS + jacobia_(:,k2)*(E(k2a,:)+[O1(k2a,:) dr.ghx(k2a,:)*H O2(k2a,:)]);
|
|
|
|
|
|
|
|
if i == M_.maximum_endo_lead
|
|
|
|
break
|
|
|
|
end
|
|
|
|
|
|
|
|
kk = find(kstate(:,2) == M_.maximum_endo_lag+i+1);
|
|
|
|
gu = dr.ghx*Gu;
|
|
|
|
[nrGu,ncGu] = size(Gu);
|
|
|
|
G1 = A_times_B_kronecker_C(dr.ghxx,Gu);
|
|
|
|
G2 = A_times_B_kronecker_C(hxx,Gu);
|
|
|
|
guu = dr.ghx*Guu+G1;
|
|
|
|
Gu = hx*Gu;
|
|
|
|
Guu = hx*Guu;
|
|
|
|
Guu(end-npred+1:end,:) = Guu(end-npred+1:end,:) + G2;
|
|
|
|
H = E1 + hx*H;
|
|
|
|
end
|
|
|
|
RHS = RHS*M_.Sigma_e(:);
|
|
|
|
dr.fuu = RHS;
|
|
|
|
%RHS = -RHS-dr.fbias;
|
|
|
|
RHS = -RHS;
|
|
|
|
dr.ghs2 = LHS\RHS;
|
|
|
|
|
|
|
|
% deterministic exogenous variables
|
|
|
|
if M_.exo_det_nbr > 0
|
|
|
|
hud = dr.ghud{1}(nstatic+1:nstatic+npred,:);
|
|
|
|
zud=[zeros(np,M_.exo_det_nbr);dr.ghud{1};gx(:,1:npred)*hud;zeros(M_.exo_nbr,M_.exo_det_nbr);eye(M_.exo_det_nbr)];
|
|
|
|
R1 = hessian*kron(zx,zud);
|
|
|
|
dr.ghxud = cell(M_.exo_det_length,1);
|
|
|
|
kf = [M_.endo_nbr-nyf+1:M_.endo_nbr];
|
|
|
|
kp = nstatic+[1:npred];
|
|
|
|
dr.ghxud{1} = -M1*(R1+f1*dr.ghxx(kf,:)*kron(dr.ghx(kp,:),dr.ghud{1}(kp,:)));
|
|
|
|
Eud = eye(M_.exo_det_nbr);
|
|
|
|
for i = 2:M_.exo_det_length
|
|
|
|
hudi = dr.ghud{i}(kp,:);
|
|
|
|
zudi=[zeros(np,M_.exo_det_nbr);dr.ghud{i};gx(:,1:npred)*hudi;zeros(M_.exo_nbr+M_.exo_det_nbr,M_.exo_det_nbr)];
|
|
|
|
R2 = hessian*kron(zx,zudi);
|
|
|
|
dr.ghxud{i} = -M2*(dr.ghxud{i-1}(kf,:)*kron(hx,Eud)+dr.ghxx(kf,:)*kron(dr.ghx(kp,:),dr.ghud{i}(kp,:)))-M1*R2;
|
|
|
|
end
|
|
|
|
R1 = hessian*kron(zu,zud);
|
|
|
|
dr.ghudud = cell(M_.exo_det_length,1);
|
|
|
|
kf = [M_.endo_nbr-nyf+1:M_.endo_nbr];
|
|
|
|
|
|
|
|
dr.ghuud{1} = -M1*(R1+f1*dr.ghxx(kf,:)*kron(dr.ghu(kp,:),dr.ghud{1}(kp,:)));
|
|
|
|
Eud = eye(M_.exo_det_nbr);
|
|
|
|
for i = 2:M_.exo_det_length
|
|
|
|
hudi = dr.ghud{i}(kp,:);
|
|
|
|
zudi=[zeros(np,M_.exo_det_nbr);dr.ghud{i};gx(:,1:npred)*hudi;zeros(M_.exo_nbr+M_.exo_det_nbr,M_.exo_det_nbr)];
|
|
|
|
R2 = hessian*kron(zu,zudi);
|
|
|
|
dr.ghuud{i} = -M2*dr.ghxud{i-1}(kf,:)*kron(hu,Eud)-M1*R2;
|
|
|
|
end
|
|
|
|
R1 = hessian*kron(zud,zud);
|
|
|
|
dr.ghudud = cell(M_.exo_det_length,M_.exo_det_length);
|
|
|
|
dr.ghudud{1,1} = -M1*R1-M2*dr.ghxx(kf,:)*kron(hud,hud);
|
|
|
|
for i = 2:M_.exo_det_length
|
|
|
|
hudi = dr.ghud{i}(nstatic+1:nstatic+npred,:);
|
|
|
|
zudi=[zeros(np,M_.exo_det_nbr);dr.ghud{i};gx(:,1:npred)*hudi+dr.ghud{i-1}(kf,:);zeros(M_.exo_nbr+M_.exo_det_nbr,M_.exo_det_nbr)];
|
|
|
|
R2 = hessian*kron(zudi,zudi);
|
|
|
|
dr.ghudud{i,i} = -M2*(dr.ghudud{i-1,i-1}(kf,:)+...
|
|
|
|
2*dr.ghxud{i-1}(kf,:)*kron(hudi,Eud) ...
|
|
|
|
+dr.ghxx(kf,:)*kron(hudi,hudi))-M1*R2;
|
|
|
|
R2 = hessian*kron(zud,zudi);
|
|
|
|
dr.ghudud{1,i} = -M2*(dr.ghxud{i-1}(kf,:)*kron(hud,Eud)+...
|
|
|
|
dr.ghxx(kf,:)*kron(hud,hudi))...
|
|
|
|
-M1*R2;
|
|
|
|
for j=2:i-1
|
|
|
|
hudj = dr.ghud{j}(kp,:);
|
|
|
|
zudj=[zeros(np,M_.exo_det_nbr);dr.ghud{j};gx(:,1:npred)*hudj;zeros(M_.exo_nbr+M_.exo_det_nbr,M_.exo_det_nbr)];
|
|
|
|
R2 = hessian*kron(zudj,zudi);
|
|
|
|
dr.ghudud{j,i} = -M2*(dr.ghudud{j-1,i-1}(kf,:)+dr.ghxud{j-1}(kf,:)* ...
|
|
|
|
kron(hudi,Eud)+dr.ghxud{i-1}(kf,:)* ...
|
|
|
|
kron(hudj,Eud)+dr.ghxx(kf,:)*kron(hudj,hudi))-M1*R2;
|
|
|
|
end
|
|
|
|
|
|
|
|
end
|
|
|
|
end
|